sat math questions i got wrong
if the circle shown has center (h,k) and contains endpoints (4,0) and (20,0) with a radius of 10, what is k?
(4-h)² + k² = 100 (20-h)² + k² = 100 (4-h)² + k² = (20-h)² + k² (4-h)² = (20-h)² 16-8h+h² = 400-40h+h² 16-8h = 400-40h -384 = -32h h = 12 (x-12)² + k² = 100 (20-12)² + k² = 100 36 = k² k= 6
let f(x) = 2x-1 . if 3f(a) = 15, what is the value of f(2a)
3(2a-1) = 15 2a-1 = 5 2a = 6 a= 3 f(6) = 2(6) - 1 f(6) = 11
30-60-90 triangle
30: 1 60: √3 90: 2
logan bought 36 pieces of bubble gum, which was 40% of the store's stock. how many pieces of bubble gum are still at the store?
36 is 40% of 90 so there were 90 pieces to begin with logan bought 36 already, so now there are 54
x³(x²-5) = -4x find x
4, 1 or 0
At a snack bar, each medium drink costs 1.85 and each large drink costs c more dollars than a medium drink. if 5 medium drinks and 5 large drinks cost a total of $20.5 what is the value of c?
5(1.85m)+5(1.85m+c)= 20.5 9.25+9.25+5c = 20.5 18.5 + 5c = 20.5 5c = 2 c = 0.4
5/x²+6x+8 = (A/x+2) + (B/x+4) what is the value of A+B if A=2.5
A(x+4)+B(x+2) = 5 2.5(x+4) + B(x+2) = 5 2.5x+10 + B(x+2) = 5 B(x+2) = -2.5x-5 B(x+2) = -2.5(x+2) B= -2.5 A+B = 2.5-2.5=0
an exponential relationship will always have
a percentage
in the ostwald process, 4 molecules of ammonia and 5 molecules of oxygen react to produce 4 molecules of nitric oxide and 6 molecules of water. which equation correctly models the relationship between number of molecules of nitric oxide and molecules of water A) 2n=3w B) 3n=2w C) 5n=4w D) 4n=5w
anytime a question asks this, you have to boil it down to simple terms 4n=6w 2n=3w the answer may seem like A) but this type of question asks for the plugged in values, not normal equation because 2(3)=3(2) the answer is 3n=2w, or B)
a triangle has area x²-x-2 and a base length of x+1. what is the height in terms of x?`
area = 0.5(b*h) x²-x-2 = 0.5(x+1)(height) 2x²-2x-4 = (x+1)(height) using synthetic division to divide the two terms, the result is 2(x+2)
On its opening day, a car dealership had an inventory of 29 cars. During the first 6 months, 18 additional cars were purchased by the dealership each week, and the sales team sold an average of 15 cars per week. During the first 6 months, 18 additional cars were purchased by the dealership each week, and the sales team sold an average of 15 cars per week. During the first 6 months, which of the following equations best models the car inventory
because the cars gained 18 and lost 5, they gained 3 y= 3x + 29
to calculate distance from a velocity time graph
calculate the area
absolute value of a function
can't be negative, so if a question is asking to set an absolute function to a negative number, there is no solution
if sin x = a
cos (90-x) = a
for positive real number x, where x^8 = 2, what is x^24
cube both sides x^24 = 8
y= a(x-h)^2 + k
displays the vertex (h,k) as constants
2 (x-1.5)² + 2(y+0.5)² = 50 what is the radius
divide both sides by 2 (x-1.5)² + (y+0.5)² = 25 radius is 5 when both sides are multiplied by 2, take into simple terms and divide both sides by 2
if a polynomial can be factored
don't use the completing square method, just factor like you normally would
in inscribed shapes
draw a line to each vertex and divide 360 by the divisors look for parallel angles from parallel lines
students in a science lab are working in groups to build both a small and large electrical circuit. A large circuit uses 4 resistors and 2 capacitors, and a small circuit uses 3 resistors and 1 capacitor. There are 100 resistors and 70 capacitors available, and each group must have enough resistors and capacitors to make one large and one small circuit. What is the maximum number of groups that could work on this lab project?
each group needs 7 resistors and 3 capacitors and there are 100 resistors and 70 capacitors available that means 100/7 and 70/3, the number of resistors will run out at 14 groups and capacitors at 23 groups. because 14 is the lower number, go with that one.
vertex is
halfway between the x-intercepts
arc length =
radius * central angle (radians)
3√f⁶a k² what is equivalent
remember that x^1/2 = sqrt x this function would be f²a k²/³
whenever a survey takes place and is asked how it would be applicable to the real population
remember to convert the sample size to population
if a question states the exponential change from one year to the next is y= 52(1.02)^t and is asked to change the years, t, to quarter-years
since dividing a year into 4 is a quarter, the new exponent would be t/4 y= 52(1.02)^(t/4)
a semicircle has diameter 6x, and two triangles have a hypotenuse 5x, what is the perimeter of the whole figure? A) 5x+3pix B) 5x+6pix C) 10x+3pix D) 10x+6pix
since it is a semicircle, the circumference will be half of what it already is the circumference of a circle is diameter*pi so the diameter here is 6pix. since it is a semicircle, the new circumference is 3pix. since there are two triangles the perimeter will be 10x. so the answer is 10x+3pix
y=a(x-2)(x+4) a is a nonzero constant. The graph of the equation in the xy=plane is a parabola with vertex (c,d). What is equal to d?
since the vertex is halfway between the x-intercepts, and the x-intercepts are 2 and -4, the vertex will be at x=-1, now we can plug this into the equation y=a(-1-2)(-1+4) y=a(-3)(3) y=-9a which means the vertex is (-1,-9a)
solution set means
solutions to both systems
let the function f be defined by f(x)=x^2+18. if a is a positive number such that f(2a) = 3f(a), what is the value of a?
step 1. plug in a and multiply it by 3 f(2a) = 3(a^2+18) step 2. plug in 2a 4a^2+18 = 3a^2+54 a^2 = 36 a=6
if (x+2)²=4, what is the solution for x?
take the square root of both sides x+2=2 x=0 x+2=-2 x=-4
if a line intersects a parabola at exactly one point
that one point is either x= the x coordinate of vertex or y= the y coordinate of vertex
drink sales in july: 1525 manilla (16 oz) 3200 manilla (24 oz) m mocha (16 oz) 175 mocha (24 oz) s espresso (16 oz) 4500 espresso (24 oz) total 16 oz= 3000 total 24 oz= 7875 if 16 oz cans represented 20% of the total number of espresso cans sold, how many 16oz cans of mocha sold?
the 16oz cans for mocha were s, and m represented 20% of the total mocha sold. so s=0.2(s+4500) 0.8s= 900 s=1125 so 1525+1125+m=3000 m=350 meaning that there were 350 cans of mocha sold
in a 45-45-90 triangle, if the 45 angled side is 5√2
the 90 degree side is (5√2)(√2) = 10
in a 30-60-90 triangle, if the 30 angled side is 5√3
the 90-degree side is 10√3 the 60-degree side is (5√3)(√3) = 15
the parabola with equation y=(x-11)² intersects the line with equation y=25 at two points, A and B. What is the length of AB?
the answer must be twice the square root of 25
higher standard deviation
the higher the degree of spread around the center of the data
adam's school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, w, that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of w for which it would be faster for Adam to walk to school? A) w-5<20 B) w-5>20 C) w+5<20 D) w+5>20
the objective of this problem is to find scenarios where the time Adam waits for the school bus and rides the school bus is greater than the time he walks to school. In other words, you want the waiting time to be greater than 15 so that the total bus time allots for greater than 20 minutes (the time it takes him to walk to school). the only option that fits this scenario is D).
a 100 percent change would occur in a quantity if
the quantity doubled
The lengths of two sides of a triangle are 30 centimeters and 35 centimeters, respectively. Between what values, in centimeters, must the length of the third side lie?
the triangle inequality theorem states that the third side of a triangle can be no less than the difference of the first two and no more than the sum of the first two. this triangle has to be between 5 to 65 centimeters
a(x-p)(x-q) displays
the x-intercepts as constants
f(x) = (x-3)(x+2) means
the xintercepts are at 3 and -2
y= ax^2 + bx + c displays
the y-intercept c as a constant
if a line is tangent to a circle
then the line is perpendicular to the radius drawn to the point of tangency
what is function x²-2x-15 simplified
there is a graph given for this function, which makes it easily identifiable that the vertex is at x=1, which makes the y-value y=-16 to solve this function algebraically, x²-2x=15 x²-2x+1 = 15+1 (x-1)² = 16 (x-1)² - 16
a science teacher is preparing the 5 stations of a science laboratory. Each station will have either experiment A materials or experiment B materials, but not both. Experiment A requires 6 teaspoons of salt, Experiment B requires 4 teaspoons of salt. If x is the number of stations that will be set up for Experiment A and the remaining will be for Experiment B, which of the following expressions represents the total number of teaspoons of salt required? A) 5x B) 10x C) 2x+20 D) 10x+20
think of the stations in terms of x A: x B: 5-x now multiply the conditionals by the number of teaspoons required 6x + 4(5-x) 6x + 20 - 4x 2x + 20 how can i remember to solve this problem? remember that, in a linear inequality, one conditional will equal x and the other will be total-x.
the previous quantity was 45 the new quantity is 18 what is the percent difference?
to find the percent difference (q1-q2/ q1) * 100 so 45-18/45 * 100 = 60
2.4x - 1.5y = 0.3 1.6x + 0.5y = -1.3 what is the x-coordinate for the solution to the system
to find the solution to the system use either elimination or substitution to use substitution first multiply the 2nd equation by 3 4.8x + 1.5y = -3.9 2.4x -1.5y = 0.3 7.2x = -3.6 x= -0.5
michael tossed a dime that landed on heads 53 times and tails 47 times. what percent of the tosses landed on heads?
to find the total probability, divide the desired / total in this case the desired are 53 and total are 100 53 / 100 = 53 percent
if a study encapsultes new york city. the results are applicable
to new york city ONLY
(a^7) (a^-4) =
a^3
the equation (24x²+25x-47)/ax-2 = -8x-3 - (53/ax-2) what is the value of a?
add 53/ax-2 to both sides 24x²+25x+6/ax-2 = -8x-3 24x²+25x+6 = (-8x-3)(ax-2) 24x²+25x+6 = -8ax²+16x-3ax+6 24x²+9x = -8ax²-3ax 24x+9 = -8ax-3a 24x+9 = a(-8x-3) a= -3
either this or that probability
add the two outcomes
2ax -15 = 3(x + 5) + 5(x -1) In the equation above, a is a constant. If no value of x satisfies the equation, what is the value of a?
after doing the algebra, the final result comes out to be: 2ax-15 = 8x+25 because no value of x satisfies the equation, this means that 2ax=8x, meaning that a=4
n²+1 / -2n+8 = -13 what are two solutions of the equation?
n²+1 = 26n-104 n²-26n+105 = 0 -5 * -21 = 105 (n-5)(n-21) = 105 n=21 or n=5
if the event of something happening is 75% and the odds of another event following it is 2:1, what is the probability that both events will occur?
odds is favorable/unfavorable while probability is favorable/total, so while the odds are 2/1, the probability is 2/3 3/4 * 2/3 = 1/2, so the probability is 0.5
in the eighth trial, how many more problems did participant 1 answer correctly than participant 2, as a percentage of the number of problems participant 2 answered correctly?
participant 1 answered 60 percent participant 2 answered 50 percent 60/50= 1.2 = 20 percent as a fraction of p2 means the percent applies to p2
angle = 45 arc= pi what is radius
pi = pi/4 * radius pi/(pi/4) = 4 radius =4
the function f is defined by f(r) = (r-4)(r+1)² if f(h-3) = 0, what is h
plug in (h-3) (h-3-4)(h-3+1)² (h-7)(h-2)² h=7 or h=2
the function f has the property that, for all x, 3f(x)=f(3x). if f(6)=12, what is the value of f(2)
plug in 2 into x and you get 3f(2) = f(6) 3f(2) = 12 f(2) = 4
if f(x) = 5x² -3 and f(x+a) = 5x²+30x+42 what is a?
plug in x+a into the first equation 5(x+a)²-30 = 5x²-30x+42 5(x²+2ax+a²) -3 = 5x²-30x+42 5x²+10ax+5a²-3=5x²+30x+42 5a²-3 = 42 5a² = 45 a² = 9 a=3 steps for solving: 1. plug in (x+a) into the original equation 2. distribute 3. find common terms and set them equal to each other
is the point (2,1) a solution for the following system of inequalities? y<4-x y< 3x-6
plugging (2,1) in both inequalities, you get 1 < 4-2 1< 2 this is a true statement 1 < 6-6 1 < 0 this is not a true statement this point is not a solution
if you have a value and are trying to find the output
put in the value as x and find the output as g(x)
equation p is defined by (p²-1)(p+2)² what are all the roots of the polynomial
p²= 1 p = 1 p = -1 p=-2 since this is a squaring problem and not a root problem, you would include both -1 and 1
y<-15x + 3000 y<5x what is the maximum point of y for solution point (x,y)?
whenever two inequalities are given and a solution point is asked, set them up to each other 5x < -15x+3000 20x < 3000 x < 150 y < 750 so the maximum point is 749
x intercepts are in relation to y
where the y-intercepts or f(x) = 0
y= 3x²-14x y= x what is the intersection of the two lines
x = 3x² - 14x 0 = 3x² - 15x 0 = 3x (x - 5) x-5 = 0 x = 5
-x < 40
x > 40
14ab + 2b/2b =
14ab/2b + 2b/2b= 14ab/2b + 1= 7a + 1 you can always take out terms in numerator and denominator
percent difference
(q1-q2/ q1) * 100
arc length=
(radius)(angle) arc = ra
(x³-9x)/(x²-2x-3) is equal to
(x(x²-9))/(x²-2x-3) (x(x+3)(x-3))/(x-3)(x+1) x(x+3)/(x+1)
the graph of x²-4x+y²+6y-24 has what radius
(x²-4x)+(y²+6y) = 24 (x²-4x+4)+(y²+6y+9) =37 (x-2)² + (y+3)² = 37
a^-5 =
1/ a^5
Sven sees a rocking horse for sale $150. The store is offering a 25% discount on everything in the store. There is a 10% sales tax that applies to the discounted price of any item in the store. How much does Sven pay for the rocking horse?
150(0.75) = 112.5 112.5(1.1) = 123.75
xiggis formula
2(s1s2)/s1+s2
a car traveled 10 miles at an average speed of 30 miles per hour and then traveled the next 10 miles at 50 miles per hour. what is the average speed
2(s1s2)/s1+s2 2(30)(50)/80 = 37.5 miles per horu
twice as many x as y
2y = x
y=x²+3x-7 y=5x-8 how many solutions exist for these quantities
5x-8 = x²+3x-7 x²-2x+1 do b²-4ac (2)²-4(1)(1) 4-4 = 0 1 solution
if a sector of a circle with a central angle of 60 has an area of 24pi, what is the radius of the circle?
60/360 = 24pi/x cross multiply to get 144pi, meaning the radius is 12
how to factor the expression 6x²-7x-3
6x²-7x-3 find two terms of -18 that equal -7. try -9 and 2. now plug that in 6x²-9x+2x-3 factor by grouping to get 3x(2x-3) + 1(2x-3)= (3x+1)(2x-3)
marta has 7,500 pesos she will convert to US dollars. 1 peso = 0.075 dollars. the exchange service will charge marta a 2% fee. how many dollars will marta receive?
7500 * 0.075 = 562.5 there is a 2% reduction so the total price will be 551.25
Odds vs. Probability
Odds: ratio of event/event not happening Probability: ratio of event/total number of events
(a^2)(a^3) =
a^2+3 = a^5
A pool initially contains 1,385 cubic feet of water. A pump begins emptying the water at a constant rate of 20 cubic feet per minute. Which function defines the volume, in cubic feet, of water in the pool t minutes after pumping begins.
f(x) = 1385 - 20t
the boiling point of water at sea level is 212 degrees. For every 1000 feet above sea level, the boiling point drops 1.84F. What equation describes this scenario
f(x) = 212 - (1.84/1000)x
solve a function with 4 terms by
factoring by grouping
An instrument shows the number of revolutions per minute made by each tire of a car. In each revolution, the car travels a distance equal to the circumference of one of its tires. The circumference of each tire is equal to 2πr, where r is the radius of the tire. If the radius of each tire on Maria's car is 0.30 meter, what is the approximate speed of Maria's car, to the nearest kilometer per hour, when the instrument is showing 779 revolutions per minute?
first convert the number of revolutions per minute to revolutions per hour
if there is a hexagon with sides length a, and the area of the hexagon is 384√3 what is the a?
first divide the hexagon into 6 triangles that all have the angle 60 degrees. then divide those 6 into 12 triangles that are 30, 60, 90 triangles. the area of each of those triangles is 32√3 because 384/12=32 because the area of a triangle is 1/2 bh, and this is a 30-60-90 triangle, the base is a/2 and height √3a/2 so 0.5(a/2)(√3a/2) = 32√3 divide root 3 from both sides (a/2)(a/2)=64 a^2/4 = 64 a^2 = 256 a=16
given equation y= -7(x+1)(x-4) what is the vertex
first find the x-coordinate of the vertex, this will be halway between intercepts -1 and 4 x = 1.5 now plug that in (1.5, 43.75)
(4x + 4)(ax - 1) - x² + 4 In the expression above, a is a constant. If the expression is equivalent to bx, where b is a constant, what is the value of b?
first, do the algebra 4ax²-4x+4ax-4-x²+4 = bx 4ax²-4x+4ax-x² = bx since there is no mention of the x² term, the x² must eliminate 4ax², meaning both are equal x² = 4ax² a = 1/4 now that the x squared terms are gone, the equation is: -4x+4ax = bx -4x + 4(1/4)x = bx -3x = bx b = -3
Jennifer bought a box of Crunchy Grain cereal. The nutrition facts on the box state that a serving size of the cereal is 0.75 cup and provides 210 calories, 50 of which are calories from fat. In addition, each serving of the cereal provides 180 milligrams of potassium, which is 5% of the daily allowance for adults. On Tuesday, Jennifer will mix Crunchy Grain cereal with Super Grain cereal for her breakfast. Super Grain cereal provides 240 calories per cup. If the total number of calories in one cup of Jennifer's mixture is 270, how much Super Grain cereal is in one cup of the mixture?
first, if the crunchy grain cereal contains 0.75 cup of 210 calories, that means that one full cup is 280 calories. Jennifer will mix a fraction of this 280 calorie cereal with a fraction of the 240 calorie cereal to create a 270 calorie cereal. both of the fractions add up to 1 because jennifer is consuming one cup. 280x+240y = 270 x+y=1
the average of 15 teachers is 40 years. if 6 additional teachers are added, the average age is 44 years. what is the average age of the 6 additional teachers?
if the average of 15 teachers is 40 years, the sum of all the teacher's ages is 600 years. say x is the sum of the 6 teachers. by adding 600+x/21 = 44 you can determine the sum of the 21 teachers is 924 years. subtract that by 600 and you get 324 years. because there are 6 of these teachers, divide 324/6 to get 54 years.
no solution one solution two solution
if the square root is positive for a binomial, or the discriminant is positive, then there are 2 solutions if the square is negative, or the discriminant is negative, there aren't any solutions if the square is 0, or the discriminant is 0, there is only 1 solution
if a chemistry problem uses the word 'unlimited'
ignore it
x+1 = 2/x+1 what is x+1
ignore the x+1 and use y instead y = 2/y y² = 2 y = √2
if h(x) has an y-intercept at d, where d is positive, which equation defines h(x)
in this equation, plugging in x should get you d this equation may look like d(3)^x as plugging in 0 for x would get you d(1)=d
any number raised to -1/2
is 1/sqrt(number)
if a is 20 percent more than b
it is NOT the same thing as b*0.8 = a instead, set up the equation 1.2b=a to set up this sort of equation, consider that the increased percent*lower variable = greater variable
Jonas is traveling by bus to visit a friend who lives 300 miles away. The friend has asked Jonas to call at least 30 minutes before arriving, so he can pick up Jonas. Jonas's bus travels at a constant speed of 45 miles per hour. Which inequality shows the number of travel hours, t, before which Jonas should call his friend?
jonas starts with 300 miles away from his friend and decreases with 45 miles per hour 300 - 45x jonas's friend wants him to call when he is 30 minutes away, or half an hour. we know that he travels 45 miles per hour, so at a half hour he'd be at 22.5 miles 300 - 45x = 22.5 x= 6.16
the dots on graphs
likely represent their data
y=3 y=ax² + b in the system of equations above, a and b are constants. For which of the following values of a and b does the system of equations have exactly two solutions? A) a=-2 b=2 B) a=-2 b=4 C) a=2 b=4 D) a=4 b=3
to solve this function, you need to keep the number and x separate a) 3= -2x²+2 2x² = -1 this system has no solution because you can't take the negative root of a number b) 3= -2x² + 4 -1 = -2x² 0.5 = x² this is positive, so it may be correct c) 3= 2x²+4 2x² = -1 not positive, incorrect d) 3= 4x²+3 4x²= 0 this only results in ONE solution, so this can't be right answer is b
what is the only scenario where 3f(x)=f(3x)
when there is a line that passes through the origin
parabola d in the xy-plane has equation x-2y^2-8y-11=0 write an equation that shows the x-intercepts of the parabola as constants or coefficients
y(0) = x-intercept x(0) = y-intercept if you want the x-intercepts as a constant, plug in 0 for y. plugging in 0 here would get you 11. so an equation with 11 as a constant or coefficient must be the answer
quarters to years in exponential ratio
y= (1.102)^(q/4) NOTE: you divide the quarters into 4
if the line y=c intersects y=-x^2+5x what is the value of c
y= -x^2+5x 0 = -(x^2-5x+25/4) -25/4 = -(x-2.5)^2 y= -(x-2.5)^2 + 25/4 y= 25/4
vertex form
y= a(x-b)² + c vertex is (b,c) if the a was negative, the vertex would still be (b,c)
√x + √9 = √64
√x + 3 = 8 √x = 5 x = 25
